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I am reading the book Complex Analysis by Gamelin and I am having trouble understanding the following definition:

Let $p_j$ be a sequence of non zero complex numbers. We say that $\prod p_j$ converges if $p_j\to 1$ and $\sum \log{p_j}$ converges (Here $\log$ denotes de principal branch of the logarithm).

I don't understand why the expression $\sum \log{p_j}$ makes sense because the $p_j$ can be negative integers, say $p_1=-1$, in this case how we define $\log(-1)$? (the principal branch of the logarithm is not defined for negative integers)

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    $\begingroup$ Notice that, however, the $p_j$-s will eventually be all in a neighbourhood of $1$. Since the first few terms are inconsequent, you may just consider any tailing sum, without worrying too much to whether $\log(-1)$ has been defined as $i\pi$ or $-i\pi$. $\endgroup$
    – user562983
    Commented Apr 19, 2019 at 21:02
  • $\begingroup$ You are right, thanks. $\endgroup$
    – Itachi
    Commented Apr 19, 2019 at 21:11

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The principal branch of the logarithm is defined by $z=re^{i\theta}\mapsto \log(r)+i\theta$, where we understand $\log(r)$ as a function on the positive reals and we force $\theta\in(-\pi,\pi]$. Since $-1=e^{i\pi}$ we find that $\log(-1)=i\pi$ (at least for the principal branch).

Thus, you can make sense of all negative integers, and you can then talk about the convergence of that sum.

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  • $\begingroup$ The problem with that is that the logarithm you are defining is not continuous, because $\log(-1+0i)=i\pi$ while $\log(-1-i0)=-i\pi$ $\endgroup$
    – Itachi
    Commented Apr 19, 2019 at 21:10
  • $\begingroup$ @Sebitas The principal branch of the logarithm isn't supposed to be a continuous function, so all is well. $\endgroup$
    – ItsJustLogsBro
    Commented Apr 19, 2019 at 21:11
  • $\begingroup$ Yes, now I understand a lot of things, thanks. Nice name btw $\endgroup$
    – Itachi
    Commented Apr 19, 2019 at 21:13

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